I will describe everything that I know about the obstacle problem in the case of a divergence form operator with bounded, uniformly elliptic coefficients. As a corollary, I will obtain a complete proof of a mean value theorem stated by Caffarelli for supersolutions to equations with such operators. In order to study the regularity of the free boundary, I will need to assume that the coefficients belong to VMO, and I will show a measure theoretic version of the alternative proven by Caffarelli in 1977. On the other hand, I will give a counter-example to show that even under the VMO assumption, blowup limits are not unique. All of this will be joint work with my student Zheng Hao.
Speaker: Ivan Blank, Department of Mathematics, Kansas State University